CHE465/865, 6-3, Lecture 1, 7 nd Sep., 6 Gouy-Chapman model (191) The double layer s not as compact as n Helmholtz rgd layer. Consder thermal motons of ons: Tendency to ncrease the entropy and make the layer dffuse (whle electrostatc attracton tends to keep ons close to nterface and make the layer more rgd). etal Soluton Dstrbuton of ons: determned by electrostatcs and stat. mechancs Assumptons: Ions: pont ons, hghly moble Solvent: delectrc contnuum etal: perfect conductor σ x Dstrbuton of ons wth dstance x from nterface: apply Boltzmann statstcs zf n = n exp n : partcle densty (#partcles/cm3) Regons of large ϕ ( x) S ( ϕ ( x) ϕ ) RT : reduced densty of postve ons, enhanced densty of anons (compared to bulk).
Charge densty nvolvng all onc speces: ( ) ρ x = n z e wth e = 1.6 1 zf = n z e exp S ( ϕ ( x) ϕ ) RT Galvan potental n soluton obeys Posson s equaton: d ϕ dx ( x) ρ ( x) = Resultng expressons s the Posson-Boltzmann equaton: d ϕ dx ( x) 1 z Fϕ = n z q x = x εε εε S exp, wher e ϕ ( ) ϕ ( ) ϕ RT Note: Ths equaton s precsely equal to the Debye-Hückel theory of onc nteractons n dlute solutons, whch determnes the charge dstrbuton around a central on. -19 C Next: Let s consder a z-z-electrolyte (*) ρ ( x) ( ) ϕ ( ) zfϕ x zf x = n zq exp exp RT RT
Debye-Hückel approxmaton Let s further assume that the potental varaton s small, zf ϕ RT ( x) << 1.e. potental varatons are ϕ << 5 mv at room-t. Ths wellknown approxmaton corresponds to the so-called Debye-Hückel approxmaton. If we use ths approxmaton n the charge dstrbuton (*) of the z-z electrolyte and then nsert ths charge dstrbuton nto the Posson-Boltzmann equaton, then we arrve at the lnear Posson-Boltzmann equaton: d ϕ dx = κ ϕ Here, κ s the nverse Debye-length, ( zf ) 1/ c 1 κ = = εε RT L D, c : bulk electrolyte concentraton (far away from nterface) κ or ts nverse L D are mportant characterstcs of the electrolyte. Debye-length L D : quanttatve measure of wdth of space charge regon wthn whch ϕ ( x) decreases from ϕ to tremendous mportance n electrochemstry and bology. S ϕ
The followng effects are mportant to remember: Larger electrolyte concentraton c smaller.a.w: the double layer becomes less dffuse Hgher temperature larger L D.a.w.: the double layer becomes more dffuse L D Potental and charge dstrbuton n the electrolyte soluton are gven n Debye-Hückel approxmaton by: σ ϕ = εε κ ( x) exp ( κ x) ( x) = exp ( x) ρ σ κ κ.e. they are exponental functons of the dstance from the nterface,.e. the excess surface charge densty σ on the metal s balanced by an exponentally decayng space-charge layer n soluton. The double layer capacty n ths approxmaton s gven by C d,dh εε = εε κ = L.e. t s gven by the plate capactor formula wth the Debye length as the effectve plate separaton. D
Table: Debye length at varous electrolyte concentratons (1-1 electrolyte) c / mol l -1 1-4 1-3 1-1 -1 L D / Å 34 96 3.4 9.6 Hgher concentratons: steeper potental drop n soluton narrower space charge regon, smaller L D hgher double layer capactes, C d,dh In practce: The Debye-Hückel (DH) )approxmaton s not vald at large electrolyte concentraton. It works well as long as the potental varaton does not exceed ϕ 8 mv.
General Case: Nonlnear Posson-Boltzmann-equaton For a z-z-electrolyte wth concentraton c the Posson- Boltzmann equaton has an explct soluton. Ths wll be added as an appendx. Here only the man results wll be gven. Potental dstrbuton n soluton etal Soluton ϕ (nterface) ϕ, = ϕ - ϕ S ϕ S (bulk) σ x The relaton between potental and charge densty of the dffuse layer s gven by dϕ 1/ zfϕ, = = ( 8RT c ) snh RT σ εε εε dx x= Dfferentatng ths expresson gves the dfferental capacty n the Gouy-Chapman model: σ εε zfϕ, d,gc = = cosh, where ϕ, = pzc ϕ, LD RT C E E
The varaton of ths capactance wth electrode potental E s thus gven by C d,gc Ths model predcts correctly the mnmum n capacty at the pzc. For large E, t predcts an unlmted rse of the capacty. E-E pzc Why would the DL capacty rse unlmtedly n ths approach? Whch assumpton s responsble for ths unphyscal behavour? It s assumed that ons are pont charges, whch could approach the electrode surface arbtrarly close wth ncreasng E. Ths leads to a very small charge separaton (consder the plate capactor as an analogue!). The capacty rses wthout lmt. What s the soluton out of ths capacty catastrophe??? Ions have a fnte sze! Arbtrarly close approach s not possble. Ions are stopped from approachng the electrode at dstances that correspond to ther rad. A further refnement of the double layer models takes ths effect nto account.
The Stern odel (194) accounts for fnte sze of ons combnes the Helmholtz and Gouy- Chapmann models two parts of double layer: (a) compact layer ( rgd layer ) of ons at dstance of closes approach (b) dffuse layer. The compact layer, x < x H, s chargefree (lnear varaton of potental)! etal σ compact layer x H dffuse layer Soluton athematcally: two capactors n seres wth total capacty 1 1 1 C C C x εε H = + = + L εε cosh z d dff,h dff,gc, Far from E pzc (large ϕ, = E Epzc ): D Fϕ RT Cdff,H << Cdff,GC Cd Cdff,H Helmholtz, rgd Close to E pzc (small ϕ, = E Epzc ): Cdff,H >> Cdff,GC Cd Cdff,GC Gouy-Chapman, dffuse Remember: the smaller guy always wns (well, not always, but here)!
Effect of electrolyte concentraton: smaller electrolyte concentraton smaller κ (L D larger), double s more dffuse! C dff,gc becomes more mportant The Stern model reproduces gross features of real systems. Potental dstrbuton n nterfacal layer and capacty varaton wth E are shown n the fgures below. What happens upon ncreasng the electrolyte concentraton? Potental dstrbuton Double layer capactance charge free moble ons
Further refnement: Grahame model (1947) Some ons (usually anons) loose hydraton shell smaller rad! Ions of smaller rad could approach the electrode closer. Dstngush three dfferent regons! Inner Helmholtz plane (IHP): through centers of small, partally solvated ons Outer Helmholtz plane (OHP): through centers of fully solvated ons Outsde OHP: dffuse layer
Ths s the best model so far. It shows good correspondence to the expermental data. Why are the potental dstrbutons (n (b)) dfferent for postve and negatve E? Ths s due to a dfference between anons and catons. Anons have a less rgd solvaton shell. They become more easly desolvated. Thus, desolvated anons could form the IHP. Ths happens at postve E. Catons keep ther solvaton shell. They, thus, cannot approach the electrode closer than to the OHP.
Appendx: Electrocapllary measurements Only applcable to lqud electrodes (metals), based on measurement of surface tenson (Lppmann s method). The method was developed specfcally for mercury electrodes. Ths s a null-pont technque,.e. the measurement of a physcal property s performed when the system has reached an equlbrum state. Such measurements are rather accurate. Balanced forces: Surface tenson of mercury n the capllary counterbalances the force of gravty π r γ cosθ = π r ρ hg c c Hg
r c s the capllary radus, γ s the surface tenson, θ s the contact angle h s the heght of the capllary column of Hg. In the measurement, the contact angle θ s measured wth a mcroscope. Above relaton then provdes values of the surface tenson γ. As the expermental parameter, the electrode potental E can be vared. What happens upon changng E? When the system s balanced (reachng a new equlbrum mnmum of Gbbs free energy!), the change n surface energy s exactly balanced by the electrcal work, A γ = QdE s d where A s s the surface area and Q s the excess charge on the surface. Therefore, the charge densty on the Hg electrode s gven by Q dγ σ = =. As de It s, thus, possble to measure the surface charge as a functon of E.